Were the forces of nature combined in one unifying force at the time of the Big Bang?
By which symmetry is this unification governed?
Are there any evidence for such unification of forces?
Has ever been published Theory or experiment in this issue? (Even original researches or unpublished theories. Anything that you can start with.)
The unifying theories of the three non-gravitational forces are known as GUT, Grand Unified Theories. The only known consistent unification of all forces including gravity (plus the known matter species) is string/M-theory.
The relevant energies or temperatures where the unification become crystal clear are very high – de facto experimentally inaccessible – which is why the new aspects of these theories were only relevant shortly after the Big Bang and why it seems impossible to make "direct experiment tests" of any grand unifying theories or more ambitious ones. However, there are some general possible experimental discoveries that, if they would emerge, would be strong arguments in favor of the validity of the frameworks.
But there exists a powerful body of theoretical evidence that string theory is right and it seems more likely than not that grand unification is realized in Nature, too (string theory admits models without grand unification of the 3 non-gravitational forces in the usual sense).
There are many isolated detailed grand unified models and we don't know which is right. Analogously, string theory has many solutions how the empty space may look – although they're connected within the string theory framework – and no one quite knows which theory/solution is the relevant one for the Universe around us.
I am afraid it makes no sense to give you more details about the theories because it seems that you have never heard anything about them and too much data would just confuse you. The literature on the subjects is huge, tens of thousands of papers, and one can't really summarize and there's no point of doing so. There are also dozens or hundreds of popular books on the subject. You would have to ask more specific questions.